A083263 Numbers k such that the difference of the largest and smallest prime factors of k divides k.

6, 12, 18, 24, 30, 36, 48, 54, 60, 70, 72, 90, 96, 108, 120, 140, 144, 150, 162, 180, 192, 198, 210, 216, 240, 270, 280, 286, 288, 300, 324, 350, 360, 384, 396, 420, 432, 450, 480, 486, 490, 510, 540, 560, 572, 576, 594, 600, 630, 646, 648, 700, 720, 750, 768

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Solutions to x mod (A006530(x) - A020639(x)) = 0.

Example

Every number k of the form 2^i * 3^j * m is a term because 3 - 2 = 1 is always a divisor of k.
Every number k of the form 2 * p * (p+2) * m is a term if p and p+2 form a twin prime pair.
Other terms include some in which the difference d = gpf(k) - lpf(k) > 2 is prime (e.g., 30 = 2*3*5 = 3*10; d = 5 - 2 = 3) and some in which it is composite (e.g., 8710 = 2*5*13*67 = 65*134; d = 67 - 2 = 65).
All terms are even. - Jon E. Schoenfield, Jul 10 2018

Mathematica

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; mi[x_] := Min[ba[x]] Do[s=ma[ba[n]]-mi[ba[n]]; If[Mod[n, s]==0, Print[{n, ba[n], s}]], {n, 1, 10000}]

Crossrefs

Cf. A033845, A071141, A006530, A020639.

Sequence in context: A085129 A236240 A242650 * A194385 A043369 A028436

Adjacent sequences: A083260 A083261 A083262 * A083264 A083265 A083266

Keyword

nonn

Author

Labos Elemer, May 12 2003

Extensions

Edited by Jon E. Schoenfield, Jul 10 2018

Status

approved